Fundamental Black–Scholes Model, Fractional Binary Approximation, No-Arbitrage, and Completeness

Sottinen (2001) constructed a binary model approximating the Black–Scholes (B-S) model driven by fractional Brownian motion using its Donsker’s type approximation. He further proved that his so-called fractional binary model has arbitrage opportunities. Torres and Tudor (2009) extended his works by approximating the B-S model driven by the Rosenblatt process, and recently, Cheng et al. (2022) extended the latter works using the Hermite process which is a generalization of the Rosenblatt process, in particular, the fractional Brownian motion. We noticed that the authors of both extended works followed the same Sottinen’s financial objective: constructing an approximating binary model which has arbitrage opportunities just like the limiting model itself, i.e., the fractional B-S model. In our paper, following a similar methodology, we rather propose a refinement of the original Sottinen’s works that ensures absence of arbitrage as follows. We construct an extension of the classical B-S model where the evolution of the risky asset is rather driven by the fundamental martingale of fractional Brownian motion introduced by Norros et al. (1999). We call it the fundamental B-S model, and we prove first that, unlike the fractional B-S model, this model is arbitrage-free and complete. We therefore compute the price of a European call option generated by this model. Next, we construct a discrete-time (fractional) binary approximation of our such model based on the Słomiński–Ziemkiewicz’s approximation of the fundamental martingale. Again, we find that, unlike Sottinen’s fractional binary market model, arbitrage can be excluded from this fractional binary approximating model, which is, hence, complete. We name it the fundamental binary market model. The Hurst index H in the interval 1/2,1 and the market prices of risk prevailing in the fundamental B-S model are key quantities in this analysis of arbitrage and market completeness. Finally, we sketch the extension of our such works to the full Hurst index range 0,1, and we conclude with some research perspective and by highlighting further recommendations for the use of our works (the second model we propose) compared to the inspiring above works.